Linear Algebra Answers

Check whether the vector (2root3, 2 ) is equally inclined to the vectors (2, 2root3) and (4,0)

Check whether the vector (2root3, 2 ) is equally inclined to the vectors (2, 2root3) and (4,0)

Check whether the vector (2root3, 2 ) is equally inclined to the vectors (2, 2root3) and (4,0)

which of the following statements are true and which are false ? justify your answer with a short proof or a counterexample. i) r^2 has infinitely many non zero, proper vector subspaces. ii) if t:v -> w is one-one linear transformation between two finite dimensional vector spaces v and w then t is invertible. iii) if a^k = 0 for a square matrix a, then all the eigen values of a are non zero. iv) every unitary operator is invertible. v) every system of homogeneous linear equations has a non zero solution.

define t : r^3>r^3 by t(x,y,x)=(x+y,y,2x-2y+2z) check that t satisfies the polynomial (x-1)^2(x-2). find the minimal polynomial of t.

Check whether the vector (2root3, 2 ) is equally inclined to the vectors (2, 2root3) and (4,0)

Consider the linear operator T:C3→C3, defined by T(z1,z2,z3)=(z1−iz2,iz1+2z2+iz3,−iz2+z3). i) Compute T∗ and check whether T is self-adjoint. ii) Check whether T is unitary.

Consider the linear operator T : C
3 → C
3
, defined by
T (z1,z2,z3) = (z1 −iz2,iz1 +2z2 +iz3,−iz2 +z3).
i) Compute T
∗
and check whether T is self-adjoint.
ii) Check whether T is unitary.

Let T : R
3 → R
3 be defined by
T (x1, x2, x3) = (x1 −x3, x2 −x3, x1).
Is T invertible? If yes, find a rule for T
−1
like the one which defines T.

Complete the set S = {x
3 +x
2 +1, x
2 +x+1, x+1} to get a basis of P3